Recent progress in studies of cobalt-based quasi-1-dimensional quantum magnets

Abstract

The interplay of crystal electric field, temperature, and spin–orbit coupling can yield a Kramer ion and thus an effective S = ½ ground state for {\mathrm{C}\mathrm{o}}^{2+} ions (3{\mathrm{d}}^7), which is often the case for low-dimensional materials. This is because a highly anisotropic structural motif can force the spins to point either “up” or “down,” thereby creating a system where spins communicate via Ising interactions. Cobalt-based quasi-1-dimensional materials have been studied in this context since the latter half of the 20th century. However, due to the development of modern characterization techniques and advances in sample preparation, the exotic physical phenomena that have generated the most interest have only emerged in the past three to four decades. This topical review mainly summarizes progress in cobalt-based quasi-1-dimensional quantum magnets and comments on a few research directions of potential future interest.

Graphical abstract

Keywords

quasi-1-dimensional materials / quantum magnet / ${\color{khaki}{{\mathrm{Co}}^{2+}}} $ ion / Ising interaction / quantum phase transition / field-induced magnetic transition / magnetic frustration / quantum fluctuations

1 Introduction

Cobalt-based quasi-1-dimensional (quasi-1D) materials show complex magnetism in the quantum regime, and thus have attracted enduring interest in materials physics. Their properties can emerge from the interplay of the Kramer ion nature of Co²⁺ ions (3d⁷) and the quasi-1D-chain structural motif of the materials. To best understand the exotic phenomena they display, one must start from the basics. The magnetic Co²⁺ ion (3d⁷) in the lattice is forced to point either “up” or “down” (effective spin-½) along the easy axis by a strong single-ion anisotropy, hence it can become a two-level system that allows spins to communicate via Ising interactions [1, 2]. CoO₆ octahedra are connected by edge-sharing or face-sharing within the 1D chains, resulting in shorter distances and smaller Co–O–Co bond angles between adjacent Co centers compared to the more commonly seen corner-sharing linkage. In some cases, these 1D chains also arrange themselves into two-dimensional (2D) lattices (e.g., the triangular lattice [3]) that can enable geometric frustration. Therefore, the dimensionality of this class of materials (i.e., the relative strengths of intra-chain and inter-chain magnetic couplings) can be tuned over a wide range, owing to the fruitful choices in the way of bridging these Co-containing 1D chains together.

The competition between intra-chain and inter-chain magnetic couplings can lead to complicated ordered and/or disordered states in cobalt-based quasi-1D quantum magnets. For example, multiple magnetic transitions can sometimes be observed in temperature-dependent magnetization data (see e.g., [4, 5]), and plateaus can be observed in field-dependent magnetization data (see e.g., [6, 7]). The superposition of short- to medium-range ordering in addition to long-range ordering [5, 8], and the transition from commensurate to incommensurate phases [9, 10] also emerges from time to time. Besides these phenomena, one of the most astonishing features displayed is a quantum phase transition (QPT) in some of the materials discussed below, just as was theoretically predicted decades ago [11]. Unlike a classical phase transition driven by thermal fluctuations, QPTs are defined as a qualitative change in the ground state of a system that occurs upon modifying a non-thermal external parameter (such as magnetic field, pressure, or doping) at a temperature of zero Kelvin [12]. The 1D quantum Ising chain in a transverse magnetic field is one of the most studied examples of a continuous QPT; this is because the zero-Kelvin ground state can be extrapolated to finite temperatures, hence making it viable to test fundamental ideas in real materials and study emergent physical properties [13, 14].

Although the study of the quantum Ising chain was addressed more than 50 years ago [15], cobalt-based quasi-1D quantum magnets exhibiting rich physics have not been fully understood even up to date and remain one of the keen research topics in the field. This topical review aims to summarize attempts that were made by both experimentalists and theorists along the journey, and to describe and rationalize spectacular findings in various types of cobalt-based quasi-1D quantum magnets. Hopefully by facilitating cross-references between different materials in this category, new leads can emerge and therefore encourage further investigations regarding this research topic.

2 One-dimensional chains consisting of edge-sharing CoO₆ polyhedra

2.1 Co-containing columbites

The most famous member of the columbite family is probably CoNb₂O₆, which has been extensively studied for decades within the physics community. Its crystal structure consists of zig-zag chains running along the c axis that are made of edge-sharing oxygen octahedra containing either Co or Nb. Early studies mainly focused on the complicated ordered magnetic states of CoNb₂O₆ in the low temperature regime, using techniques including neutron scattering, specific heat capacity, magnetic susceptibility and magnetization. It is reported that the material has two intrinsic magnetic transitions, at T₁ ~ 3.0 K and T₂ ~ 1.9 K, with the former one associated with a 3D helical magnetic ordered phase while the latter one with a pronounced 2D character even in the 3D ordered state [4, 16−18]. The ferromagnetic Ising chains, containing Co²⁺, are only weakly coupled to each other as they are separated by two rows of non-magnetic NbO₆ chains [Fig.1(a)]. If J₀ denotes the intra-chain ferromagnetic exchange along the easy axis, which is in the a–c plane at an angle ±31° to the c axis [Fig.1(b)] and J₁ and J₂ denote the much weaker inter-chain antiferromagnetic interactions, then |J₁|, |J₂| \llJ₀ [Fig.1(c)] [2, 19]. Researchers have worked on the H//c–T magnetic phase diagram of CoNb₂O₆, revealing antiferromagnetic, ferrimagnetic, and incommensurate phases, plus various ordered phases that are field-induced between the ferrimagnetic and saturated paramagnetic phases [9].

CoNb₂O₆ is considered as an ideal system to study the interplay of quantum fluctuations and quantum criticality, which are two of the most intriguing topics in current condensed-matter physics. As illustrated above, the Ising chains in CoNb₂O₆ are arranged in a triangular lattice, which is archetypal for geometrically frustrated magnets, and hence quantum fluctuations can be induced in the system [1]. Thus ⁹³Nb nuclear magnetic resonance (NMR) was used to investigate the quantum spin fluctuations in CoNb₂O₆ near the quantum critical point (QCP), owing to the hyperfine interactions between Co electron spins and ⁹³Nb nuclear spins [20]. In addition, CoNb₂O₆ has been found to be a close realization of the transverse Ising magnet (TIM) in a real material. In the absence of a transverse field, the weakly coupled ferromagnetic Ising chains remain in ordered states (spins aligned “up” or “down” along the Ising axis), but when a transverse field that can overcome the exchange interactions is applied, the system undergoes a phase transition to a disordered state at a QCP, attributed to the Ising spins all lying in the quantum superposition of “up” and “down” states [1, 2]. Heat capacity results show that the spin entropy is largely enhanced at the QCP and find evidence for gapless spin excitations that are both participating in, and affected by, the QPT [2]. Anomalous lattice softening is also observed near the QCP in CoNb₂O₆, due to the relativistic spin–orbit interaction in the quantum critical regime [21]. Inelastic neutron scattering experiments have provided direct evidence for quantum criticality in the Ising chains by using strong transverse fields to tune CoNb₂O₆ through its QCP [14]. In complement to the five kink bound states observed by neutron scattering [14], high-resolution time-domain terahertz (THz) spectroscopy finds more features in the far infrared for CoNb₂O₆, with four additional kink bound states and a new higher energy excitation below the continuum resolved [22].

However, with more experimental studies about the ordered/disordered states of CoNb₂O₆ coming out, substantial deviations from the ideal transverse-field Ising model were noted. For example, domain walls in CoNb₂O₆ display quantum motion at zero applied field, which is inconsistent with a pure 1D Ising model [9, 14, 17, 23, 24]. This feature was probed by a theoretical approach, using a microscopic spin-exchange Hamiltonian to demonstrate the crucial role of glide symmetry breaking in the system [25]. Several experimental studies have also focused on the domain-wall confinement/freezing in CoNb₂O₆ [26−28]. A “twisted Kitaev chain” composed of a bond-dependent Ising interaction was proposed for CoNb₂O₆, inspired by similar interactions found in those honeycomb Kitaev spin liquids [29]. This idea was recently backed up by theorists based on the JKΓ model [Heisenberg (J)–Kitaev (K)–Gamma (Γ)] for a 3d⁷ systems such as Co²⁺. They mapped out a microscopic origin for Ising behavior in spin–orbit coupled 1D chains and posit CoNb₂O₆ as a rare Kitaev chain [30].

CoTa₂O₆ [Fig.2(a)] and CoV₂O₆ [Fig.2(b, c)] have also been studied, but not to a comparable extent so far. There are very limited papers on CoTa₂O₆ [31−33], but one study did grow a single crystal of this compound by using the optical floating-zone technique [34]. Its cation-ordered rutile-derived structure results in a Co–Ta–Ta–Co sequence within each 1D chain, which might dilute the intra-chain Co–Co magnetic coupling to a certain extent and forfeit some exotic phenomena. Generally speaking, CoV₂O₆ has received more attention from the community than CoTa₂O₆. It has two structural forms – a low-temperature phase [triclinic γ-CoV₂O₆, Fig.2(b)] and a high-temperature phase [monoclinic α-CoV₂O₆, Fig.2(c)], separated by a structural transition which occurs at 953 K [35]. Upon resolving the magnetic structure from neutron powder diffraction data, it was found that the γ phase contains two independent Co sites which are not perfectly aligned but show angles different from 180°, and thus cannot unambiguously show ferromagnetic order inside the chains [36]. In contrast, the magnetic moments in the α phase lie in the ac plane and are coupled within the chains running along the b axis, no matter in the ground or field-induced states [37−40]. Attempts to grow CoV₂O₆ single crystals have been made by using both the flux [35, 41] and the floating-zone techniques [42]. Very recently, however, quantum annealing resulting from time-reversal symmetry breaking in a tiny transverse field was reported in a single-crystal study of α-CoV₂O₆. The many-body simulations in that work point out that a tiny applied transverse field can profoundly enhance quantum spin fluctuations within the system [43]. Thus, α-CoV₂O₆ could be analogous to its sister compound CoNb₂O₆ in a certain way, with a potential superiority – the easement in the strength of the applied transverse field. The above-mentioned results in the literature suggest that exotic physical phenomena can be seen in CoV₂O₆ as well, as long as high-quality samples can be synthesized. In other words, obstacles such as the competition between polymorphs or the volatility of starting reagents during the crystal growth process need to be overcome, hence yielding the much needed “ultra-clean” samples for further study in the quantum regime.

2.2 Co-containing pyroxenes

Pyroxenes, with a general formula AMX₂O₆ (A = mono- or di-valent cation, M = transition metal cation, X = tetravalent cation), are one of the most abundant minerals in Earth’s crust. In its crystal lattice (Fig.3), the edge-sharing MO₆ octahedra form infinite 1D chains that propagate in a zigzag fashion along the c axis. Adjacent chains are bridged by XO₄ tetrahedra along the a axis, while they are non-linked along the b axis because of the A cations residing in the cavities. This quasi-1D system can result in substantial anisotropic physical properties, hence serving as a potential platform to study in detail the emergent quantum phenomena that originate from the interplay of intra- and inter-chain couplings. The long-range antiferromagnetic unit cell of Co-containing pyroxenes was initially obtained from neutron diffraction experiments performed on the polycrystalline powder samples [44, 45]. However, they were then put on the back burner because nothing particularly interesting was observed under low magnetic fields. In contrast, the analogous Cr, Mn and Fe materials have been in the spotlight due to the observed multiferroicity and magnetoelectric effects, etc. [46−51]. In more recent years, the Co-containing pyroxenes have regained attention from the community due to the emergent anisotropic field-induced (meta)magnetic transitions that arise from the joint effect brought by Co²⁺ ions in this highly versatile crystal structure [6, 7, 52, 53].

CoGeO₃, a simpler version of the ACoX₂O₆, pyroxene, has been studied in depth. Large CoGeO₃ single crystals were grown in a high pressure mirror furnace. The direction-dependent magnetic characterization performed on the single crystal reveals highly anisotropic magnetic susceptibility in the system [52]. In addition, if the external magnetic field was applied along the chain direction (i.e., the c axis), unusually well-defined 1/3 plateaus emerged in the field-dependent magnetization data, despite the absence of an apparent triangular lattice in the structural motif [6] (Fig.4). These studies suggest that the magnetic interactions between spins in these weakly coupled Co 1D chains are more complicated than originally anticipated, hence worth further investigation.

The magnetic properties of the SrCoGe₂O₆ polycrystalline powder samples were studied in detail using a combination of neutron scattering, heat capacity, abinitio methods, and linear spin-wave theory [53]. This work demonstrates a bond-dependent Kitaev exchange model for this material, together with a field-induced state achieved through sabotaging the fragile antiferromagnetic ordering between spins. What’s more, it also proposes that more intriguing phenomena might emerge once the dimensionality of the system (i.e., relative coupling strength of intra- and inter-chain interactions) has been further manipulated [53]. Following this lead, the most straightforward way to modify the dimensionality of the system is to vary the XO₄ bridging unit between Co 1D chains, i.e., substituting Ge with Si in the inter-chain super-exchange pathway. A systematic study of the CaCoGe_2−xSi_xO₆ series was carried out, with both end members CaCoGe₂O₆ and CaCoSi₂O₆ studied in depth by growing macroscopic single crystals [7]. On cooling below the Néel temperatures, a sharp field-induced transition in magnetization is observed for CaCoGe₂O₆, while multiple magnetization plateaus beneath the full saturation moment are spotted for CaCoSi₂O₆. These contrasting behaviors potentially arise from the different electron configurations of Ge and Si, in which the 3d orbitals are filled in the former but empty in the latter, enabling electron hopping. Thus, SiO₄ can aid the inter-chain super-exchange pathway between Co²⁺ centers while GeO₄ tends to block it during magnetization. These (meta)magnetic transitions were found to be highly anisotropic, and the external magnetic field required to induce them strongly depends on the choice of bridging unit XO₄ (Fig.5). In addition, heat capacity data collected on all these Co-containing pyroxenes revel an effective spin-½ for Co(II) ions in 1D chains, even after being manipulated by high magnetic fields [7, 53].

These recent studies have demonstrated that Co-containing pyroxenes can be considered as a new playground for Kitaev physics. The highly tunable dimensionality and the fruitful choice of elements within this family of materials opens more possibilities to study exotic phases with fractionalized excitations, as well as advance our understanding in Kitaev materials in general [6, 7, 52, 53].

2.3 A(II)Co₂V₂O₈ and its derivatives

Though the 1D chain in A(II)Co₂V₂O₈ family still consists of edge-sharing CoO₆ octahedra, it distinguishes itself from the two families discussed above by adopting a slightly different magnetic structure. Among this family, the A = Ba/Sr versions [Fig.6(a)] have been extensively studied during the past two decades. Spins within the screw chains of Co²⁺ rotating around the four-fold c axis are strongly antiferromagnetically coupled, while the inter-chain coupling is much weaker [Fig.6(b)] [54, 55]. This eventually yields a long-range antiferromagnetic order at substantially low temperatures (~ a few Kelvins). Thus, the A(II)Co₂V₂O₈ family possesses a strong magnetic anisotropy, owing to the quasi-1D nature of their magnetic structure, and becomes effective 1D spin-1/2 antiferromagnets with Heisenberg–Ising (or XXZ) exchange anisotropy due to the Co²⁺ ions in the lattice [56]. Although the screw chains in (Ba/Sr)Co₂V₂O₈ are antiferromagnetic, unlike the zig-zag ferromagnetic chains found in CoNb₂O₆, they are still rich in exotic physical phenomena including but not limited to quantum phase transitions.

Inelastic neutron scattering and terahertz spectroscopy reveal the existence of spinon confinement in (Ba/Sr)Co₂V₂O₈. A broad spinon continuum is observed above the Néel temperature (~ 5 K) with zero magnetic field applied. When cooling below T_N, pairs of spinons are confined owing to the sizable inter-chain attractive linear potential, hence the broad continuum splits into unconventional discrete spin excitations on the so-called Zeeman ladders [57−59]. The behavior of (Ba/Sr)Co₂V₂O₈ in an applied transverse field (perpendicular to the c axis) is found to be quite intriguing as well. A substantial suppression of the antiferromagnetic order is induced by applying a transverse field along the a axis in BaCo₂V₂O₈. However, the antiferromagnetic order can be retained even in high transverse fields, if the field is applied along the ab-plane diagonal. This peculiar angular-dependent phenomenon can be rationalized by the strongly anisotropic g-tensor for Co²⁺ ions in this material, as the result of the characteristic screw chain structure [54, 56, 60, 61]. Compared to CoNb₂O₆, the 1D transverse-field Ising universality has not been illustrated in (Ba/Sr)Co₂V₂O₈ until recently. Two QPTs in SrCo₂V₂O₈ are revealed in the ultra-low temperature NMR measurements, with the first QPT attributed to the rapid suppression of the antiferromagnetic order and the second one featured with gapless excitations [62]. By combining neutron diffraction and inelastic neutron scattering experiments, terahertz spectroscopy and NMR, as well as theoretical analyses, the profile of quantum phase transitions is mapped out in detail for BaCo₂V₂O₈, accompanied by the observation of the diagnostic E₈ particles [55, 63−67]. The longitudinal magnetic field applied along the easy axis (approximately along the c axis) further enriches the physics of this exotic family. Upon the elevated strength of an applied field, the commensurate antiferromagnetic ground state is destabilized into an incommensurate longitudinal spin density wave phase through a first-order transition, and then turns into a transverse canted antiferromagnet for both BaCo₂V₂O₈ [10, 68] and SrCo₂V₂O₈ [69]. What’s more spectacular is the experimental realization of Bethe strings – complex bound states of magnetic excitations in a condensed-matter system: after it was theoretically predicted by Von Bethe almost a century ago [70]. High-resolution terahertz spectroscopy [71, 72] and inelastic neutron scattering [73] resolve these string states in SrCo₂V₂O₈ under a strong longitudinal field.

The divalent A cation does not necessarily have to be picked up from the alkaline group, for example, Pb²⁺ can also reside in those cavities. Although PbCo₂V₂O₈ has been preliminarily investigated in the past [74, 75], its detailed magnetic structure and phase diagram has not been probed until recently. A series of complex (quantum) phase transitions are discovered, regarding both the magnitude and direction of the applied magnetic field [76]. The Pb version seems to exhibit the potential for properties that could be generally comparable to (Ba/Sr)Co₂V₂O₈ and thus is worth further investigation. Compositions containing other divalent A cations with tolerated ionic radii, and other pentavalent cations like Ta⁵⁺ or Nb⁵⁺ in place of V⁵⁺ could be the subject of future study as well.

3 One-dimensional chains consisting of face-sharing CoO₆ polyhedra

3.1 The A_n+2Co_n+1O_3n+3 (n = 1) family

Derived from the 2H-perovskite related oxides, Ca₃Co₂O₆ and its derivatives constitute a non-neglectable sub-class of cobalt-based quasi-1D quantum magnets and have been most intensively studied in the A_n+2Co_n+1O_3n+3 family. The crystal structure of Ca₃Co₂O₆ (space group R-3c) has been solved by powder X-ray and neutron diffraction data [77]. Ca₃Co₂O₆ distinguishes itself from other types of Co-based chain materials by the alternating arrangement of face-sharing CoO₆ trigonal prisms (TP) and CoO₆ octahedra (Oh) within the 1D chains running along the c axis [Fig.7(a)]. These chains form a triangular lattice in the ab plane [Fig.7(b)] [78].

There are two types of Co centers – one resides in the trigonal prism (Co_trig) and the other resides in the octahedron (Co_oct). In order to understand the magnetism of Ca₃Co₂O₆, the oxidation state and electronic configuration of each type of Co center need to be mapped out first. Early experimental and theoretical studies made several suggestions, among which the two most agreed combinations are – low-spin (S = 0) trivalent Co_oct and high-spin (S = 2) trivalent Co_trig, or low-spin (S = 1/2) tetravalent Co_oct and high-spin (S = 3/2) divalent Co_trig [79−86]. After studying this compound in detail by using X-ray photoemission spectroscopy, X-ray absorption and magnetic circular dichroism, powder and single-crystal neutron diffraction, etc., trivalent Co centers in both coordination environments (low-spin Co_oct and high-spin Co_trig) have generally become the mainstream understanding within the community [8, 87, 88].

The magnetic ground state of Ca₃Co₂O₆ is preliminarily depicted as a triangular lattice of ferromagnetic 1D chains (spins coupled across the spin-bearing Co_trig sites within the chains), with the much weaker inter-chain coupling described by the antiferromagnetic Ising model. An incommensurate magnetic ground state is suggested by a few studies [89−91]. In addition to a modulated partially disordered antiferromagnetic structure along the c axis, the coexistence of a shorter-range order in the ab plane has also been unveiled [8]. This argument has been backed up by a magnetocaloric study of the metamagnetic transitions present in the system [92]. Field-induced plateaus are spotted in the magnetization curves of Ca₃Co₂O₆, which is generally rare but not rare among the subjects of this review. But what makes Ca₃Co₂O₆ particularly interesting is when cooling below 10 K, the plateau at 1/3 of the saturation moment (observed at the temperature range of ~ 10–25 K) further splits into equidistant sub-steps, signaling quantum tunneling of the magnetization. The sub-steps are clearer when a faster sweep rate is employed [8, 81, 82, 93−96]. The origin of this intriguing phenomenon (two steps above 10 K and at least four steps below 10 K), i.e., the magnetic-order dynamics in Ca₃Co₂O₆ has attracted enduring interest from the community during the past two decades. Monte Carlo and Wang−Landau simulations have been carried out by theorists. Their calculation results can generally reproduce the experimental observations in quantitative agreement at different sweep rates of the applied magnetic fields. They also point out that nonequilibrium magnetization and magnetization inhomogeneity are the key factors needed to induce the four-step state below 10 K [97−103]. Formation [104] and deformation [105] of spin-density-wave microphases in Ca₃Co₂O₆ are investigated as well. Other than magnetic field, external factors such as time [106, 107], pressure [108] and A-site chemical doping [83] have been introduced to study their influence on the magnetic order in Ca₃Co₂O₆. What’s more, several other techniques such as orbital imaging [109], NMR [110], muon-spin spectroscopy [111] have also been used to study Ca₃Co₂O₆. In summary, the nature of the exotic magnetic behavior of Ca₃Co₂O₆ remains an ongoing puzzle and worth further investigations.

The fact that the crystal structure of Ca₃Co₂O₆ has two distinguished B-cation sites provides an opportunity to mix Co with other transition metals. Thus, its derivatives have also been studied by the community. The Mn-doping of Ca₃Co₂O₆ has drawn some attention. The discovery of collinear-magnetism-driven ferroelectricity in Ca₃CoMnO₆ [Fig.8(a)] shines a spotlight on this isostructural sister compound [112]. By employing ab initio electronic structure calculations and X-ray absorption spectroscopy, Co²⁺ and Mn⁴⁺ adopt an ordered arrangement, with the former in trigonal prisms and the latter in octahedra, both in high-spin states [113]. The magnetic-order dynamics in the Ca₃Co_2−xMn_xO₆ series is also examined. Similar magnetization behavior is found at much higher magnetic field sweep rates than that in Ca₃Co₂O₆ – six orders of magnitude faster. In addition to magnetization, steps are also spotted in the magnetostriction, electric polarization, and magnetocaloric effect, potentially proving the rich physics in this material [114]. A partially disordered antiferromagnetic phase, in Ca₃CoRhO₆ [Fig.8(b)], has also been evidenced by a neutron diffraction study [115]. The oxidation states of Co and Rh have caused controversy. A spin-polarized electronic band structure calculations study suggests both cations in their trivalent states [116], while an X-ray photoemission spectroscopy study leans towards a combination of Co²⁺ and Rh⁴⁺ [87]. The electronic structure and/or transport properties of Ca₃CoNiO₆ and Ca₃CoIrO₆ have also been briefly studied [87, 117].

3.2 The A_n+2Co_n+1O_3n+3 (n = 2) family

By inserting an additional octahedron into the repeating unit (1TP-1Oh) of A_n+2Co_n+1O_3n+3 (n = 1), the 1D chains running along the c axis alternate in a 1TP-2Oh manner and yield a general formula of A_n+2Co_n+1O_3n+3 (n = 2). The simplest version, namely A₄Co₃O₉ (A = Ca, Sr, Ba), is known for its thermoelectric properties [118]. Evidence for two-dimensional antiferromagnetic order in this system has been obtained by muon spin rotation and relaxation (μ⁺SR) techniques [111, 119, 120]. However, in terms of magnetism, the Mn-doped analogues ACo_3−xMn_xO₉ remain the main subject of study in this sub-field. The solubility of Mn in A₄Co₃O₉ is determined to be in the range of 0.5 ≤ x ≤ 2 [121]. To avoid any potential structural disorder between Co²⁺ and Mn⁴⁺ sites, the community is particularly interested in compositions with Co²⁺:Mn⁴⁺ = 1:2 [122, 123]. Thus, Co²⁺ and Mn⁴⁺ can adopt an ordered arrangement, with the former in trigonal prisms and the latter in octahedra [Fig.9(a, b)]. Sr_4−yCa_yCoMn₂O₉ oxides (y = 0 and 2) are found to be an ideal playground for studying the interplay among single-ion magnetism (SIM), single-chain magnetism (SCM), and long-range order (LRO) phenomena, which are usually discussed in coordination polymers or complexes [124]. The y = 2 member has received additional attention due to its deviation from a textbook geometrically frustrated triangular lattice from the structural point of view – there are three subclasses of chains shifted from each other so some of the intrachain distances between spins differ from others. This is reflected in its dc magnetization data [Fig.9(c)], two transitions denoted as T_N and T^* are observed in a very narrow temperature range (25–35 K). This double-peak feature is then further investigated by ac magnetization over 14 frequencies between 10⁻¹ and 10⁴ Hz [Fig.9(d)], revealing that a peculiar pre-transitional short-range ordering is present in the system [5]. Some other dopants like M = Zn, Cu and Rh are also explored for the A_n+2Co_n+1O_3n+3 (n = 2) family [125, 126].

4 Other Co-based quasi-1-dimensional quantum magnets

4.1 Co²⁺-containing garnets

Garnets with general composition R₃B₂(AO₄)₃ (R, B and A represent different individual cationic sites) have cubic symmetry and consist of a 3D network of BO₆ octahedra that share corners with bridging AO₄ tetrahedra. However, the garnet structure cannot be easily visualized in terms of close packed ionic spheres like those in perovskites or spinels, instead, it can be described as rods out of the octahedral sites that are then joined together by vacant triangular prisms. Therefore, it is only natural to anticipate that magnetic ions with strong single-ion anisotropy, such as Co²⁺, can potentially lower the dimensionality of the magnetic unit cell of garnets. Consistent with this concept, by using low-temperature powder neutron diffraction, spins in CaY₂Co₂Ge₃O₁₂ are found to adopt an unusual anisotropic and chain-like antiferromagnetic arrangement [127]. Besides, two Co²⁺ containing garnets, CaY₂Co₂Ge₃O₁₂ and NaCa₂Co₂V₃O₁₂, are reported to exhibit magnetic-field-induced dielectric anomalies [128]. Garnets can accommodate fruitful choices of elements, and thus Co²⁺-containing garnets might be a class of materials that have been overlooked in terms of low-dimensional magnetism due to its apparent highly symmetric cubic crystal structure.

4.2 Co²⁺-containing halide compounds

CsCoX₃ (X = Cl and Br) are reported to exhibit exotic phenomena that are analogous to some of the oxides discussed above in this review. Raman scattering measurements have been performed to study the spin dynamics in this system [129−132]. The spin-wave response corresponding to an S = ½ Ising-like chain antiferromagnet is unveiled by inelastic neutron scattering experiments [133], as well as the domain walls [134, 135] previously predicted by Villain [136] and soliton response occurring at elevated temperatures [137]. Theorists have also made multiple attempts to justify these experimental observations [138−141].

Although RbCoCl₃ has been preliminarily investigated by Fourier transform infrared spectroscopy [142] and Raman scattering [132, 143], it regains some attention from the community very recently. By combining the high-resolution neutron spectroscopy with Monte Carlo simulations, the thermal control of spin excitations in RbCoCl₃ [Fig.10(a, b)] is investigated to an unprecedented extent [144]. This work motivates a few more studies to further characterize the two magnetic phase transitions at low temperatures and the unusual behavior of the spin−phonon coupling in RbCoCl₃ [145−147].

4.3 Co²⁺-containing hybrid compounds

Hybrid compounds including both organic and inorganic building blocks could offer additional degrees of freedom when modifying the structural motif to tailor the need for target properties. The size of organic building blocks varies in a much wider range than is typically seen for inorganics, thus, the dimensionality of the Co²⁺-containing hybrid compounds could be tuned to a greater extent compared to those inorganic ones. A few examples of Co-based one-dimensional spin-chain hybrid compounds have been studied by the community, displaying the merit of the versatility of organic ligands.

For example, neutron scattering measurements along with magnetic characterizations reveal that the dominant interaction in the Co²⁺-containing hydrazinium sulfate material Co(N₂H₅)₂(SO₄)₂ [Fig.10(c)] is the intra-chain antiferromagnetic interaction. Polarized neutron powder diffraction measurements also point out that this material exhibits an anisotropic magnetization on the Co site with indications of Ising-type interactions in the ab plane, thus motivating the need for the growth of large, high-quality single crystals to further study the direction-dependent physical properties in this system [148]. In addition, the reaction of Co(NCS)₂ with 3-bromopyridine leads to the formation of various types of complexes depending on the solvent and reaction conditions used. This series of complexes show qualitatively different magnetic properties upon variation in the coordination environment of Co²⁺ ions and the overall dimensionality of materials [149]. These reported Co²⁺-containing hybrid compounds demonstrate that this is a sub-class of quasi-1D chain materials with high potential in quantum magnetism.

4.4 Co²⁺-containing chalcogenides

For quasi-1D Co²⁺-containing chalcogenides, these systems warrant a separate review, as the different bonding interactions associated with oxide and heavier chalcogenide anions lead to distinct yet equally fascinating magnetic behaviors in the quantum regime.

5 Conclusions

In this topical review, we have summarized many of the intensively studied Co-based quasi-1D materials that are known or proposed to be quantum magnets over the past few decades, ranging from quasi-1D spin-chain compounds consisting of edge-sharing and face-sharing CoO₆ polyhedra, to other Co²⁺-containing systems such as highly symmetric garnets, halide and hybrid compounds. The exotic phenomena observed either commonly or unique among the subjects of the current review, make them one of the keen research topics in both the theoretical and experimental physics world. While some of the problems have been figured out, due to the development of modern theories, characterization techniques and advances in sample preparation skills, there are still a lot of open questions under ongoing debate, affirming Co-based quasi-1D materials an active playground with huge potential for studying magnetism in the quantum regime.

Among the subjects of current review, we can find that some material systems such as pyroxenes and garnets are not even studied in an equivalent depth compared to the others, not to mention that some of the studies only came in very recently. Besides, the material systems intensively studied in other keen research areas of condensed matter physics substantially outnumber those in this one. Hence it is of priority to overcome obstacles in high-quality crystal growth and enrich the variety of material systems investigated in this field. Therefore, by writing this review, we hope to bring researchers with comprehensive understanding of both condensed-matter physics and materials chemistry on board, because this is an exciting interdisciplinary research field that should attract scientists who work at the border between physics and chemistry.

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